Showing papers on "Natural exponential family published in 1999"

Generalized Exponential Distributions

Abstract: Summary The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.

Relative loss bounds for on-line density estimation with the exponential family of distributions

Abstract: We consider on-line density estimation with a parameterized density from the exponential family. The on-line algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss which is the negative log-likelihood of the example w.r.t. the past parameter of the algorithm. An off-line algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the on-line algorithm over the total loss of the off-line algorithm. These relative loss bounds hold for an arbitrary sequence of examples. The goal is to design algorithms with the best possible relative loss bounds. We use a certain divergence to derive and analyze the algorithms. This divergence is a relative entropy between two exponential distributions.

A General Class of Exponential Inequalities for Martingales and Ratios

Abstract: In this paper we introduce a technique for obtaining exponential inequalities, with particular emphasis placed on results involving ratios. Our main applications consist of approximations to the tail probability of the ratio of a martingale over its conditional variance (or its quadratic variation for continuous martingales). We provide examples that strictly extend several of the classical exponential inequalities for sums of independent random variables and martingales. The spirit of this application is that, when going from results for sums of independent random variables to martingales, one should replace the variance by the conditional variance and the exponential of a function of the variance by the expectation of the exponential of the same function of the conditional variance. The decoupling inequalities used to attain our goal are of independent interest. They include a new exponential decoupling inequality with constraints and a sharp inequality for the probability of the intersection of a fixed number of dependent sets. Finally, we also present an exponential inequality that does not require any integrability conditions involving the ratio of the sum of conditionally symmetric variables to its sum of squares.

Exponential family state space models based on a conjugate latent process

Abstract: Summary. This paper introduces non-linear and non-Gaussian state space models with analytic updating recurNions for filtering and prediction. This new class of models involves some well-known results in the theory of exponential models and of exponential dispersion models and the latent process is defined in such a way that both the filtering and the prediction distributions turn out to be conjugate to the observation distribution at each step. The corresponding analytic and inferential properties are investigated and some simple examples are presented.

Can data recognize its parent distribution

Abstract: This study is concerned with model selection of lifetime and survival distributions arising in engineering reliability or in the medical sciences. We compare various distributions—including the gamma, Weibull, and lognormal—with a new distribution called geometric extreme exponential. Except for the lognormal distribution, the other three distributions all have the exponential distribution as special cases. A Monte Carlo simulation was performed to determine sample sizes for which survival distributions can distinguish data generated by their own families. Two methods for decision are by maximum likelihood and by Kolmogorov distance. Neither method is uniformly best. The probability of correct selection with more than one alternative shows some surprising results when the choices are close to the exponential distribution.

Can Data Recognize Its Parent Distribution

Abstract: This study is concerned with model selection of lifetime and survival distributions arising in engineering reliability or in the medical sciences. We compare various distributions, including the gamma, Weibull and lognormal, with a new distribution called geometric extreme exponential. Except for the lognormal distribution, the other three distributions all have the exponential distribution as special cases. A Monte Carlo simulation was performed to determine sample sizes for which survival distributions can distinguish data generated by their own families. Two methods for decision are by maximum likelihood and by Kolmogorov distance. Neither method is uniformly best. The probability of correct selection with more than one alternative shows some surprising results when the choices are close to the exponential distribution.

Credibility evaluation for the exponential dispersion family

Abstract: It has long been established that under regularity conditions, the linear credibility formula with an appropriate credibility factor produces exact fair premium for claims or losses whose distribution is a member of the natural exponential family. Recently, this result has been extended to a richer family of distribution, the exponential dispersion family which comprised of several distributions, some of which are heavy-tailed and as such could be of significant relevance to actuarial science. The family draws its richness from a dispersion parameter σ2=1/λ which is equal to 1 in the case of the natural exponential family. In this paper neither λ is regarded known, nor a fully specified prior distribution for λ is assumed. Instead, by establishing a link between the m.s.e. of the linear credibility and Fisher information we derive optimal credibility for the case where only the mean and variance of λ are specified.

Exponential Shake-and-Bake: theoretical basis and applications.

Abstract: The simple cosine function used in the formulation of the traditional minimal principle and the related Shake-and-Bake algorithm is here replaced by a function of exponential type and its expected value and variance are derived. These lead to the corresponding exponential minimal principle and its associated Exponential Shake-and-Bake algorithm. Recent applications of the exponential function to several protein structures within the Shake-and-Bake framework suggest that this function leads, in general, to significant improvements in the success rate (percentage of trial structures yielding solution) of the Shake-and-Bake procedure. However, only in space group P1 is it presently possible to assign optimal values a priori for the exponential-function parameters.

Characterization of Lifetime Distributions Using Measure of Uncertainty

Abstract: In the present paper, we characterize exponential, Pareto, Wei bull and beta distributions using the concept of measure of uncertainty defined by Ebrahimi and Pellery (1995). Further, a new measure of uncertainty is introduced and its properties are discussed. Finally, we present a characterization theorem for extreme value distribution by extending the concept into negative set-up.

Miscellanea. A multivariate family of distributions on (0, ∞)p

Abstract: In the context of parametric survival analysis, it is necessary to specify probability distributions on (0, ∞). Typically, the exponential, Weibull, gamma, Pareto or log-normal is used. However, attempts to generalise these distributions to a multivariate setting have proved problematic. This paper introduces a univariate family of distributions, called the beta-log-normal family, motivated by a mixture representation of some of the more typical distributions and which generalises naturally to the multivariate case.

Simulation in exponential families

Abstract: An acceptance-rejection algorithm for the simulation of random variables in statistical exponential families is described. This algorithm does not require any prior knowledge of the family, except sufficient stati stics and the value of the parameter. It allows simulation from many members of the exponential family. We present some bounds on computing time, as well as the main properties of the empirical measures of samples simulated by our methods (functional Glivenko-Cantelli and central limit theorems). This algorithm is applied in order to evaluate the distribution of M-estimators under composite alternatives; we also propose its use in Bayesian statistics in order to simulate from posterior distributions.

Tail distribution modelling using the richter and power exponential distributions

Abstract: The vast majority of HMM–based speech recognition systems u se Gaussian mixture models as the state distribution model. Th e use of these distributions is motivated more by ease of training , decoding and the fact that a sufficient number of Gaussian componen ts may be used to approximate any distribution, than some under lying aspect of the data being modelled. If distributions were selected that better modelled the observed data, fewer compon ents should be required and recognition accuracy should improve . This paper examines two distributions for improving the modelli ng of the tails of the densities. The first distribution, the Richt er distribution, fits within the general framework of Gaussian compon ent tying, but has some attractive attributes for decoding. The second distribution, the power exponential, does not fit within a tying framework. Despite gains in likelihood, indicating that th e Gaussian components are sub–optimal in a likelihood sense, only small gains in recognition performance were observed on a large vo cabulary speech recognition task.

A class of generalized Poisson distributions

Abstract: In these paper we introduce a new class of exponential sums from which various known as well as new exponential sums are generated. Then a class of generalized Poisson distributions is defined (GPDs). Some well known and new distributions are obtained from GPDs. Some recurrence relations of the parameters of these distributions is studied. The suitability of these distributions is illustrated through some real life data.

The uniform power distribution

Abstract: This paper introduces a generalization of the normal distribution: the uniform power distribution. It is a symmetric and unimodal family of distributions, defined on the real line, and is closely related to the exponential power family. The exponential power family was introduced to allow the modelling of kurtosis. The uniform power family matches the exponential power family with respect to the range of kurtosis. However, whereas the exponential is somewhat difficult to work with, the contrary is true for the uniform power family.

Estimation of Common Parameter of Several Exponential Distributions

Abstract: The problem of estimation of common parameter of k exponential distributions with known but possibly unequal coefficients of variation has been solved by showing its structural equivalence with the problem of estimation of parameter of single exponential distribution with weighted average coefficient of variation.

Rendezvous of the Poisson and Exponential Distributions at the World Cup of Soccer

Abstract: Summary Data from the World Cup provide excellent illustrations of Poisson and exponential distributions

Ruin Probability of an InsuranceCompany in the Case where Intervals Between PaymentsHave Unequal Exponential Distributions

Abstract: This paper considers the problem of finding the ruin probability of an insurance company in the case where intervals between payments have unequal exponential distributions and values of payments have identical exponential distributions. As particular cases we consider those where the values of payments are unequally distributed and have Erlang distributions, and also those where the expiration times are order statistics of a sample from the exponential distribution. The results obtained can be applied in the queueing theory.

Some properties of the expectation coordinate system of a binary exponential (distribution) family

Abstract: It has been shown that there exist monotonic relations between the elements of the normalized Fisher information matrix (correlation coefficients) and the natural parameters, if a certain inclusion relation is satisfied in a binary exponential family which is defined on n probability variables taking a binary value of either 1 or 0, and if their higher- order mutual interactions are taken into account. Due to this property, changes of correlation coefficients can be approximately predicted from the changes in the values of the parameters alone. In this paper, we study the way in which changes of the expectation parameters are related to changes in the elements of the Fisher information matrix and the normalized Fisher information matrix (negative partial correlation coefficients) in the expectation coordinate system that is dual to the natural parameter coordinate system in terms of the Fisher metric in the binary exponential family. Specifically, it can be shown that there exist monotonic properties of these statistical measures, the sufficient conditions for which are inclusion relations, in contrast to the similar relations between the correlation coefficients and natural parameters. By using this property, increases, decreases, and changes of the statistical measures can be predicted from the changes of the parameters without directly calculating the change of the measure itself. The properties of monotonic increase and decrease are also shown to be dual to each other. These properties are not direct conclusions derived from the properties in the natural coordinate system, but are newly obtained by analysis of the mixture family corresponding to the binary exponential family. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(12): 88–97, 1999

Minimax posterior regret actions for exponential families of distributions and weighted squared error loss

Abstract: The paper focuses on the choice of appropriate loss in a Bayesian framework. In particular, a criterion based on minimax posterior regret is considered. Attention is paid to classes of weighted squared error loss. A general result is presented and an application to one parameter exponentia families of distributions is given.

Estimation of the parameter of identically distributed exponential quantities

Abstract: A sequence of Independent random exponential variables with identical distribution is considered. The conditions of strong consistency for the parameter of the sequence under certain conditions are presented.

Exponential rates for the error probabilities in selection procedures

Abstract: For a sequence of statistical experiments with a finite parameter set the asymptotic behavior of the maximum risk is studied for the problem of classification into disjoint subsets. The exponential rates of the optimal decision rule is determined and expressed in terms of the normalized limit of moment generating functions of likelihood ratios. Necessary and sufficient conditions for the existence of adaptive classification rules in the sense of Rukhin [Ru1] are given. The results are applied to the problem of the selection of the best population. Exponential families are studied as a special case, and an example for the normal case is included.